Earthquakes and the Richter scale
Energy hiding behind small numbers
For students, teachers, and the simply curious. No prior knowledge of logarithms needed — if you can do 3 + 5, you can follow this to the end.
Earthquakes and the Richter scale
You've probably heard of the Richter scale on the news. A magnitude 3 earthquake you barely feel, a magnitude 7 can destroy a city. But the difference between 3 and 7 isn't "a bit more" — it's enormous.
Scientists designed the Richter scale so that each step of 1 magnitude means exactly 10 ↑ 1.5 times more energy:
10 ↑ 1.5 = 31.62310 ↑ 1.5(10 up 1.5)So from magnitude 5 to magnitude 6: 31.6× more energy. From 5 to 7? That's two steps: 31.6 × 31.6 = about 1000× more energy. The numbers on the scale look small, but the energy behind them is huge.
How much magnitude for double energy?
If a full step (+1 magnitude) means 10 ↑ 1.5 more energy, then a half step (+0.5) means:
10 ↑ (1.5 × 0.5) = 10 ↑ 0.7510 ↑ 0.75(10 up 0.75)About 5.6× more energy. In general, a step of Δ magnitude means:
energy ratio = 10 ↑ (1.5 × Δ)Now the question: for which Δ does the energy exactly double? We fill in 2:
10 ↑ (1.5 × Δ) = 2This has the shape a ↑ b = c, where 1.5 × Δ is in the position of b — the right side of ↑. From the rule card: right side unknown → use ⇓.
1.5 × Δ = 2 ⇓ 102 ⇓ 10(2 double‑down 10)That gives about 0.301. But that's 1.5 × Δ, not Δ itself. To get Δ alone, divide by 1.5:
Δ = 2 ⇓ 10 ÷ 1.52 ⇓ 10 ÷ 1.5(2 double‑down 10 divided by 1.5)About 0.2. That means a magnitude 7.2 earthquake releases roughly twice the energy of a 7.0. Let's see what that looks like:
Each tiny 0.2 step doubles the energy. From 5.0 to 6.0 is just one number on the scale, but it means 32 times more energy. The Richter scale is hiding a power relationship — and the ⇓ symbol is what lets you see into it.
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2 ↑ 3 = 82 = 8 ↓ 3left unknown? use down3 = 8 ⇓ 2right unknown? use double-down